appendix_SMcoh.tex

\section{Comparing StableMoor $\ulow$ to IMU $\uhead$}
\label{apdx:ulow}

\def\ubt{\ensuremath{\vec{u}_\mathrm{BT}}}

\begin{figure}[t]
  \centering
  \includegraphics{BT_IMU_Coherence02}
  \caption{Coherence between IMU-measured motion of StableMoor buoy and ADP bottom-track velocity for $1.0<\bar{U}<1.5$. The horizontal dashed line indicates the 95\% confidence level for the 102 spectral windows in this estimate. The vertical dotted line indicates the frequency of the high-pass filter applied to the IMU accelerometers in estimating $\uhead$.}
  \label{fig:SM_coh}
\end{figure}

To better understand the IMU's signal-to-noise ratio, we compare the motion of the StableMoor buoy from the ADP bottom track measurements, $\ubt$, to the IMU's estimates of ADP motion. To do this, we compute the IMU's estimate of ADP motion using equation \eqref{eqn:uhead}, and replacing $\ell^{*}$ with the vector that points from the IMU to the ADP head. In this case, we use a 5 minute high-pass filter ($f_a=0.00333$) in \eqref{eqn:uhead}; this reduces spectral reddening that otherwise contaminates coherence estimates and preserves the $\uacc$ estimates at the frequencies where we wish to compare to $\ubt$ (Figure \ref{fig:SM_coh}).  We also linearly interpolate the ADP measurements of $\ubt$ onto the times of the ADV-IMU measurements.

The coherence between these two signals is high and statistically significant over 1.5 decades---from 0.03 to 0.8 Hz (Figure \ref{fig:SM_coh}, \citealt[][]{Priestley1981}). The $v$ component has the highest coherence, 98\%, because this is the direction that has the most motion (i.e., these estimates have a higher signal-to-noise ratio).  The $u$ and $w$ components have a slightly lower coherence, 80\% and 65\%, respectively.

On the low-frequency side, our interpretation is that the signal-to-noise ratio of the IMU decreases dramatically below 0.03 Hz, resulting in low coherence. On the high-frequency side, Doppler noise in the ADP measurements contaminates its estimates of motion, causing the decrease in coherence at 0.8 Hz. A comparison of the phase between these signals shows that there is no lag between the measurements (not shown).

These results help to inform the selection of zero-lag filters used to estimate $\ulow$ from $\ubt$. In particular, by selecting 0.2 Hz, we target the middle of the coherence peak between the two measurements. Furthermore, the rapid decrease in coherence below 0.03 Hz provides an objective measurement of the frequency at which IMU measured velocity becomes unreliable in the flow conditions we observed. 



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